The initial technical obstruction to constructing quotients of unstable strata is clear enough: the strata consist by definition of GIT-unstable points, so their quotients are empty. Since GIT semistability depends on a choice of linearisation, a natural approach is to vary the linearisation \(\mathcal {L}\)  by twisting by a character to change the semistable locus, as is described by variation of GIT [1]}, [2]}. However, there may not exist a character \(\chi _\beta \)  of \(G\)  which we can use to make all of \(S_\beta \)  semistable (for example, if \(G=\mathrm {SL}_n\) , there are no non-trivial characters). Fortunately, if we pick a representative 1-PS \(\lambda _\beta \)  of the conjugacy class \(\beta \) , this determines a parabolic group \(P_\beta :=P(\lambda _\beta )< G\)  and locally closed subscheme \(Y_\beta ^{ss} \subset S_\beta \)  such that \(S_\beta \)  is the \(G\) -sweep of \(Y_\beta ^{ss}\) , and such that flowing under \(\lambda _\beta \)  induces a retraction \(p_\beta : Y_\beta ^{ss} \rightarrow Z_\beta ^{ss}\)  [3]}. Moreover, a categorical quotient of \(G\)  acting on \(S_\beta \)  is equivalent to a categorical quotient of \(P_\beta \)  acting on \(Y_\beta ^{ss}\) . The parabolic subgroup \(P_\beta \)  has larger character group than \(G\)  and in particular, there is a character \(\chi _\beta \)  such that the twisted linearisation \(\mathcal {L}_\beta \)  (which we refer to as a borderline linearisation below) produces a categorical \(P_\beta \) -quotient of \(Y_\beta ^{ss}\)  [4]}. However, this quotient factors by the retraction \(p_\beta \)  followed by the categorical quotient of \(Z_\beta ^{ss}\)  under the action of the Levi \(L_\beta < P_\beta \) . In particular, this categorical quotient is far from being an orbit space. Often the moduli theoretic interpretation of the retraction \(p_\beta \)  is given by taking the associated graded of a Harder–Narasimhan filtration and so the obtained categorical quotient is just a product of moduli spaces for the successive semistable quotients in the HN filtration.
